Exposure is the amount of money that you are at risk of losing on a given position (i.e. on a UST 10 year bond), portfolio of positions, strategy (selling covered calls for example), or counterparty, usually represented as a percentage of your total assets.
Interbank exposure is the exposure of banks to other banks either through owning debt or stock, or by having open positions with the other banks as counterparties.
Leveraging occurs when the value of your position is more than the value of what you are trading in. One example of this is borrowing money (i.e. creating debt for yourself) to buy bonds. The amount of your own funds that you are using to pay for the position is "leveraged" by the debt so that you are risking more than 100% of your capital if, for example, the bond became worthless). Another example would be buying futures "on margin" where you only put up the margin value of the trade and not the full cost.
The problem with these leveraged positions is what happens if a credit event (default etc.) happens. Since a large amount of the leverage is being "passed on" as banks are issuing debt to buy other banks' debt who are issuing debt to buy debt there is a risk that a single failure could cause an unravelling of these leveraged positions and, since the prices of the bonds will be falling resulting in these leveraged positions losing money, it will cause a cascade of losses and defaults. If a leveraged position becomes worth less than the amount of real (rather than borrowed or margined) money that was put up to take the position then it is almost inevitable that the firm in that position will default on the requirements for the leverage. When that firm defaults it sparks all of the firms who own that debt to go through the same problems that it did, hence the contagion.